Pairs of Finite Dimensional Nilpotent and Filiform Lie Algebras

Let (N,L) be a pair of finite dimensional nilpotent Lie algebras. If N admits a complement K in L such that dim N = n and dim K = m, then dim M(N,L) = 1/2n(n + 2m- 1) t(N,L), where M(N,L) is the Schur multiplier of the pair (N,L) and t(N,L) is a nonnegative integer. In this paper, we characterize the pair (N,L) for t(N,L)=0, 1, 2, … , 23, where N is a finite dimensional filiform Lie algebra and N,K are ideals of L such that L = N ⊕ K. Moreover, we classify the pair (N,L) for s′ (N,L) = 3, where S′ (N,L) = 1/2 (n- 1)(n -2) + 1 + (n- 1)m – dim M(N,L), L is a finite dimensional nilpotent Lie algebra and N is a nonabelian ideal of L.